It can be done, of course. You could create a user named “anonymous” for example, and then show the activity update form adding an option for posting “as ‘username’ or ‘anonymous'”. And if the last is selected, add the variable via post and add checking via backend. If everything is ok, post to the database with the “anonymous user” ID. Voila! The user posted as anonymous. 🙂

Activity component doesn’t support anonymous comments or behaviour; it wasn’t built with that in mind. Per Juan’s suggestion above, you could create a fake user account and write code to assign comments to that user, but… sounds hacky 🙂